The linear equation elimination method is a technique used to solve a system of linear equations. It involves manipulating the equations in such a way that one of the variables is eliminated, leaving a new equation with one less variable. This process is repeated until a solution is obtained for all variables. Here is an example of how to use the elimination method to solve a system of linear equations:

Suppose we have the system of equations:

2x + 3y = 10 4x – y = 14

To solve this system using the elimination method, we need to eliminate one of the variables. Let’s choose y. We can do this by multiplying the first equation by 4 and the second equation by 3, so that the coefficients of y are equal and opposite:

8x + 12y = 40 12x – 3y = 42

Now we can add the two equations together to eliminate y:

20x = 82

Solving for x, we get:

x = 82/20 = 4.1

Now we can substitute this value of x back into one of the original equations to solve for y. Let’s use the first equation:

2x + 3y = 10 2(4.1) + 3y = 10 8.2 + 3y = 10 3y = 1.8 y = 0.6

Therefore, the solution to the system is:

x = 4.1 y = 0.6

We can verify that these values satisfy both equations in the system.

The elimination method is used to solve a system of linear equations. Here are the steps:

1. Write the system of linear equations in standard form: ax + by = c
2. Choose one of the variables (x or y) and eliminate it by adding or subtracting the equations. The goal is to get the coefficients of one of the variables to cancel out.
3. Solve the resulting equation for the remaining variable.
4. Substitute this value into one of the original equations and solve for the other variable.
5. Check your solution by substituting the values into both equations.

Here’s an example:

Solve the system of linear equations using the elimination method:

2x + 3y = 7 4x + 5y = 13

1. Write the equations in standard form:

2x + 3y = 7 4x + 5y = 13

2. Choose to eliminate x. Multiply the first equation by -2 and add it to the second equation to eliminate x:

-4x – 6y = -14 4x + 5y = 13

-y = -1

3. Solve for y:

y = 1

4. Substitute y = 1 into one of the original equations and solve for x:

2x + 3y = 7 2x + 3(1) = 7 2x + 3 = 7 2x = 4 x = 2

5. Check the solution by substituting x = 2 and y = 1 into both equations:

2(2) + 3(1) = 7 4 + 3 = 7

4(2) + 5(1) = 13 8 + 5 = 13

The solution is x = 2 and y = 1.